4.2: The Elements
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Learning Objectives
- Define a chemical element and give examples of the abundance of different elements.
- Represent a chemical element with a chemical symbol.
An element is a substance that cannot be broken down into simpler chemical substances. There are about 90 naturally occurring elements known on Earth. Using technology, scientists have been able to create nearly 30 additional elements that do not occur in nature. Today, chemistry recognizes 118 elements—some of which were created an atom at a time. Figure \(\PageIndex{1}\) shows some of the chemical elements.
Abundance
The elements vary widely in abundance. In the universe as a whole, the most common element is hydrogen (about 90% of atoms), followed by helium (most of the remaining 10%). All other elements are present in relatively minuscule amounts, as far as we can detect.
| Earth’s Crust | Earth (overall) | ||
|---|---|---|---|
| Element | Percentage | Element | Percentage |
| oxygen | 46.1 | iron | 34.6 |
| silicon | 28.2 | oxygen | 29.5 |
| aluminum | 8.23 | silicon | 15.2 |
| iron | 5.53 | magnesium | 12.7 |
| calcium | 4.15 | nickel | 2.4 |
| sodium | 2.36 | sulfur | 1.9 |
| magnesium | 2.33 | all others | 3.7 |
| potassium | 2.09 | ||
| titanium | 0.565 | ||
| hydrogen | 0.14 | ||
| phosphorus | 0.105 | ||
| all others | 0.174 | ||
| Source: D. R. Lide, ed. CRC Handbook of Chemistry and Physics , 89th ed. (Boca Raton, FL: CRC Press, 2008–9), 14–17. |
On the planet Earth, however, the situation is rather different. Oxygen makes up 46.1% of the mass of Earth’s crust (the relatively thin layer of rock forming Earth’s surface), mostly in combination with other elements, while silicon makes up 28.2%. Hydrogen, the most abundant element in the universe, makes up only 0.14% of Earth’s crust. Table \(\PageIndex{1}\) lists the relative abundances of elements on Earth as a whole and in Earth’s crust. Table \(\PageIndex{2}\) lists the relative abundances of elements in the human body. If you compare Table \(\PageIndex{1}\) and Table \(\PageIndex{2}\), you will find disparities between the percentage of each element in the human body and on Earth. Oxygen has the highest percentage in both cases, but carbon, the element with the second highest percentage in the body, is relatively rare on Earth and does not even appear as a separate entry in Table \(\PageIndex{1}\); carbon is part of the 0.174% representing “other” elements. How does the human body concentrate so many apparently rare elements?
| Element | Percentage by Mass |
|---|---|
| oxygen | 61 |
| carbon | 23 |
| hydrogen | 10 |
| nitrogen | 2.6 |
| calcium | 1.4 |
| phosphorus | 1.1 |
| sulfur | 0.20 |
| potassium | 0.20 |
| sodium | 0.14 |
| chlorine | 0.12 |
| magnesium | 0.027 |
| silicon | 0.026 |
| iron | 0.006 |
| fluorine | 0.0037 |
| zinc | 0.0033 |
| all others | 0.174 |
| Source: D. R. Lide, ed. CRC Handbook of Chemistry and Physics , 89th ed. (Boca Raton, FL: CRC Press, 2008–9), 7–24. |
The relative amounts of elements in the body have less to do with their abundances on Earth than with their availability in a form we can assimilate. We obtain oxygen from the air we breathe and the water we drink. We also obtain hydrogen from water. On the other hand, although carbon is present in the atmosphere as carbon dioxide, and about 80% of the atmosphere is nitrogen, we obtain those two elements from the food we eat, not the air we breathe.
LOOKING CLOSER: PHOSPHOROUS , THE CHEMICAL BOTTLENECK
There is an element that we need more of in our bodies than is proportionately present in Earth’s crust, and this element is not easily accessible. Phosphorus makes up 1.1% of the human body but only 0.105% of Earth’s crust. We need phosphorus for our bones and teeth, and it is a crucial component of all living cells. Unlike carbon, which can be obtained from carbon dioxide through photosynthesis, there is no phosphorus compound present in our surroundings that can serve as a convenient source. Phosphorus, then, is nature’s bottleneck. Its availability limits the amount of life our planet can sustain.
Higher forms of life, such as humans, can obtain phosphorus by selecting a proper diet (plenty of protein); but lower forms of life, such as algae, must absorb it from the environment. When phosphorus-containing detergents were introduced in the 1950s, wastewater from normal household activities greatly increased the amount of phosphorus available to algae and other plant life. Lakes receiving this wastewater experienced sudden increases in growth of algae. When the algae died, concentrations of bacteria that ate the dead algae increased. Because of the large bacterial concentrations, the oxygen content of the water dropped, causing fish to die in large numbers. This process, called eutrophication , is considered a negative environmental impact.
Today, many detergents are made without phosphorus so the detrimental effects of eutrophication are minimized. You may even see statements to that effect on detergent boxes. It can be sobering to realize how much impact a single element can have on life—or the ease with which human activity can affect the environment.
Names and Symbols
Each element has a name. Some of these names date from antiquity, while others are quite new. Today, the names for new elements are proposed by their discoverers but must be approved by the International Union of Pure and Applied Chemistry, an international organization that makes recommendations concerning all kinds of chemical terminology.
Today, new elements are usually named after famous scientists.
The names of the elements can be cumbersome to write in full, especially when combined to form the names of compounds. Therefore, each element name is abbreviated as a one- or two-letter chemical symbol. By convention, the first letter of a chemical symbol is a capital letter, while the second letter (if there is one) is a lowercase letter. The first letter of the symbol is usually the first letter of the element’s name, while the second letter is some other letter from the name. Some elements have symbols that derive from earlier, mostly Latin names, so the symbols may not contain any letters from the English name. Table \(\PageIndex{3}\) lists the names and symbols of some of the most familiar elements.
| aluminum | Al | magnesium | Mg |
| argon | Ar | manganese | Mn |
| arsenic | As | mercury | Hg* |
| barium | Ba | neon | Ne |
| bismuth | Bi | nickel | Ni |
| boron | B | nitrogen | N |
| bromine | Br | oxygen | O |
| calcium | Ca | phosphorus | P |
| carbon | C | platinum | Pt |
| chlorine | Cl | potassium | K* |
| chromium | Cr | silicon | Si |
| copper | Cu* | silver | Ag* |
| fluorine | F | sodium | Na* |
| gold | Au* | strontium | Sr |
| helium | He | sulfur | S |
| hydrogen | H | tin | Sn* |
| iron | Fe | tungsten | W † |
| iodine | I | uranium | U |
| lead | Pb* | zinc | Zn |
| lithium | Li | zirconium | Zr |
| *The symbol comes from the Latin name of element. | |||
|---|---|---|---|
| † The symbol for tungsten comes from its German name— wolfram . |
Element names in languages other than English are often close to their Latin names. For example, gold is oro in Spanish and or in French (close to the Latin aurum ), tin is estaño in Spanish (compare to stannum ), lead is plomo in Spanish and plomb in French (compare to plumbum ), silver is argent in French (compare to argentum ), and iron is fer in French and hierro in Spanish (compare to ferrum ). The closeness is even more apparent in pronunciation than in spelling.
Example \(\PageIndex{1}\)
Write the chemical symbol for each element without consulting Table \(\PageIndex{3}\) "Element Names and Symbols".
- bromine
- boron
- carbon
- calcium
- gold
- Answer a
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Br
- Answer b
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B
- Answer c
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C
- Answer d
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Ca
- Answer e
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Au
Exercise \(\PageIndex{1}\)
Write the chemical symbol for each element without consulting Table \(\PageIndex{3}\).
- manganese
- magnesium
- neon
- nitrogen
- silver
- Answer a
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Mn
- Answer b
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Mg
- Answer c
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Ne
- Answer d
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N
- Answer e
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Ag
Example \(\PageIndex{2}\)
What element is represented by each chemical symbol?
- Na
- Hg
- P
- K
- I
- Answer a
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sodium
- Answer b
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mercury
- Answer c
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phosphorus
- Answer d
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potassium
- Answer e
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iodine
Exercise \(\PageIndex{2}\)
What element is represented by each chemical symbol?
- Pb
- Sn
- U
- O
- F
- Answer a
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lead
- Answer b
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tin
- Answer c
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uranium
- Answer d
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oxygen
- Answer e
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fluorine
Concept Review Exercises
- What is an element?
- Give some examples of how the abundance of elements varies.
- Why are chemical symbols so useful? What is the source of the letter(s) for a chemical symbol?
Answers
- An element is the basic chemical building block of matter; it is the simplest chemical substance.
- Elements vary from being a small percentage to more than 30% of the atoms around us.
- Chemical symbols are useful to concisely represent the elements present in a substance. The letters usually come from the name of the element.
Key Takeaways
- All matter is composed of elements.
- Chemical elements are represented by a one- or two-letter symbol.


4: "Hotspots"- Mantle Thermal Plumes
The vast majority of earthquakes and volcanic eruptions occur near plate boundaries, but there are some exceptions. For example, the Hawaiian Islands, which are entirely of volcanic origin, have formed in the middle of the Pacific Ocean more than 3,200 km from the nearest plate boundary. How do the Hawaiian Islands and other volcanoes that form in the interior of plates fit into the plate-tectonics picture?
Introduction
In 1963, J. Tuzo Wilson, the Canadian geophysicist who discovered transform faults, came up with an ingenious idea that became known as the "hotspot" theory. Wilson noted that in certain locations around the world, such as Hawaii, volcanism has been active for very long periods of time. This could only happen, he reasoned, if relatively small, long-lasting, and exceptionally hot regions -- called hotspots -- existed below the plates that would provide localized sources of high heat energy (thermal plumes) to sustain volcanism. Specifically, Wilson hypothesized that the distinctive linear shape of the Hawaiian Island-Emperor Seamounts chain resulted from the Pacific Plate moving over a deep, stationary hotspot in the mantle, located beneath the present-day position of the Island of Hawaii. Heat from this hotspot produced a persistent source of magma by partly melting the overriding Pacific Plate. The magma, which is lighter than the surrounding solid rock, then rises through the mantle and crust to erupt onto the seafloor, forming an active seamount. Over time, countless eruptions cause the seamount to grow until it finally emerges above sea level to form an island volcano. Wilson suggested that continuing plate movement eventually carries the island beyond the hotspot, cutting it off from the magma source, and volcanism ceases. As one island volcano becomes extinct, another develops over the hotspot, and the cycle is repeated. This process of volcano growth and death, over many millions of years, has left a long trail of volcanic islands and seamounts across the Pacific Ocean floor.
Space Shuttle photograph of the Hawaiian Islands, the southernmost part of the long volcanic trail of the "Hawaiian hotspot" (see text). Kauai is in the lower right corner (edge) and the Big Island of Hawaii in the upper left corner. Note the curvature of the Earth (top edge). (Photograph courtesy of NASA.)
According to Wilson's hotspot theory, the volcanoes of the Hawaiian chain should get progressively older and become more eroded the farther they travel beyond the hotspot. The oldest volcanic rocks on Kauai, the northwesternmost inhabited Hawaiian island, are about 5.5 million years old and are deeply eroded. By comparison, on the "Big Island" of Hawaii -- southeasternmost in the chain and presumably still positioned over the hotspot -- the oldest exposed rocks are less than 0.7 million years old and new volcanic rock is continually being formed.
Above: Artist's conception of the movement of the Pacific Plate over the fixed Hawaiian "Hot Spot," illustrating the formation of the Hawaiian Ridge-Emperor Seamount Chain. (Modified from a drawing provided by Maurice Krafft, Centre de Volcanologie, France). Below: J. Tuzo Wilson's original diagram (slightly modified), published in 1963, to show his proposed origin of the Hawaiian Islands. (Reproduced with permission of the Canadian Journal of Physics .)
The possibility that the Hawaiian Islands become younger to the southeast was suspected by the ancient Hawaiians, long before any scientific studies were done. During their voyages, sea-faring Hawaiians noticed the differences in erosion, soil formation, and vegetation and recognized that the islands to the northwest (Niihau and Kauai) were older than those to the southeast (Maui and Hawaii). This idea was handed down from generation to generation in the legends of Pele, the fiery Goddess of Volcanoes. Pele originally lived on Kauai. When her older sister Namakaokahai, the Goddess of the Sea, attacked her, Pele fled to the Island of Oahu. When she was forced by Namakaokahai to flee again, Pele moved southeast to Maui and finally to Hawaii, where she now lives in the Halemaumau Crater at the summit of Kilauea Volcano. The mythical flight of Pele from Kauai to Hawaii, which alludes to the eternal struggle between the growth of volcanic islands from eruptions and their later erosion by ocean waves, is consistent with geologic evidence obtained centuries later that clearly shows the islands becoming younger from northwest to southeast.
Prominent world hotspots
Although Hawaii is perhaps the best known hotspot, others are thought to exist beneath the oceans and continents. More than a hundred hotspots beneath the Earth's crust have been active during the past 10 million years. Most of these are located under plate interiors (for example, the African Plate), but some occur near diverging plate boundaries. Some are concentrated near the mid-oceanic ridge system, such as beneath Iceland, the Azores, and the Galapagos Islands.
World map showing the locations of selected prominent hotspots; those labelled are mentioned in the text. (Modified from the map This Dynamic Planet. )
A few hotspots are thought to exist below the North American Plate. Perhaps the best known is the hotspot presumed to exist under the continental crust in the region of Yellowstone National Park in northwestern Wyoming. Here are several calderas (large craters formed by the ground collapse accompanying explosive volcanism) that were produced by three gigantic eruptions during the past two million years, the most recent of which occurred about 600,000 years ago. Ash deposits from these powerful eruptions have been mapped as far away as Iowa, Missouri, Texas, and even northern Mexico. The thermal energy of the presumed Yellowstone hotspot fuels more than 10,000 hot pools and springs, geysers (like Old Faithful), and bubbling mudpots (pools of boiling mud). A large body of magma, capped by a hydrothermal system (a zone of pressurized steam and hot water), still exists beneath the caldera. Recent surveys demonstrate that parts of the Yellowstone region rise and fall by as much as 1 cm each year, indicating the area is still geologically restless. However, these measurable ground movements, which most likely reflect hydrothermal pressure changes, do not necessarily signal renewed volcanic activity in the area.
Authors' Note: Since this booklet's publication in 1996, vigorous scientific debate has ensued regarding volcanism at "hotspots." New studies suggest that hotspots are neither deep phenomena nor "fixed" in position over geologic time, as assumed in the popular plume model. See http://www.mantleplumes.org/ ."
Snow-capped 4,169-m-high Mauna Loa Volcano, Island of Hawaii, seen from the USGS Hawaiian Volcano Observatory. Built by Hawaiian hotspot volcanism, Mauna Loa -- the largest mountain in the world -- is a classic example of a shield volcano. (Photograph by Robert I. Tilling, USGS.)
Contributors and Attributions
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W. Jacquelyne Kious and Robert Tilling (" The Dynamic Earth " via the U.S. Geological Survey )


1.2.20: Adding, Subtracting, and Multiplying Radical Expressions
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Adding and Subtracting Radical Expressions
Adding radical expressions with the same index and the same radicand is just like adding like terms. We call radicals with the same index and the same radicand like radicals to remind us they work the same as like terms.
Like Radicals
Like radicals are radical expressions with the same index and the same radicand.
We add and subtract like radicals in the same way we add and subtract like terms. We know that \(3x+8x\) is \(11x\).Similarly we add \(3 \sqrt{x}+8 \sqrt{x}\) and the result is \(11 \sqrt{x}\).
Think about adding like terms with variables as you do the next few examples. When you have like radicals, you just add or subtract the coefficients. When the radicals are not like, you cannot combine the terms.
Example \(\PageIndex{1}\)
Simplify:
- \(2 \sqrt{2}-7 \sqrt{2}\)
- \(5 \sqrt[3]{y}+4 \sqrt[3]{y}\)
- \(7 \sqrt[4]{x}-2 \sqrt[4]{y}\)
Solution :
a.
\(2 \sqrt{2}-7 \sqrt{2}\)
Since the radicals are like, we subtract the coefficients.
\(-5 \sqrt{2}\)
b.
\(5 \sqrt[3]{y}+4 \sqrt[3]{y}\)
Since the radicals are like, we add the coefficients.
\(9 \sqrt[3]{y}\)
c.
\(7 \sqrt[4]{x}-2 \sqrt[4]{y}\)
The indices are the same but the radicals are different. These are not like radicals. Since the radicals are not like, we cannot subtract them.
You Try \(\PageIndex{1}\)
Simplify:
- \(5 \sqrt{3}-9 \sqrt{3}\)
- \(5 \sqrt[3]{y}+3 \sqrt[3]{y}\)
- \(5 \sqrt[4]{m}-2 \sqrt[3]{m}\)
- Answer
-
- \(-4 \sqrt{3}\)
- \(8 \sqrt[3]{y}\)
- \(5 \sqrt[4]{m}-2 \sqrt[3]{m}\)
For radicals to be like, they must have the same index and radicand. When the radicands contain more than one variable, as long as all the variables and their exponents are identical, the radicands are the same.
Example \(\PageIndex{2}\)
Simplify:
- \(2 \sqrt{5 n}-6 \sqrt{5 n}+4 \sqrt{5 n}\)
- \(\sqrt[4]{3 x y}+5 \sqrt[4]{3 x y}-4 \sqrt[4]{3 x y}\)
Solution :
a.
\(2 \sqrt{5 n}-6 \sqrt{5 n}+4 \sqrt{5 n}\)
Since the radicals are like, we combine them.
\(0 \sqrt{5 n}\)
Simplify.
\(0\)
b.
\(\sqrt[4]{3 x y}+5 \sqrt[4]{3 x y}-4 \sqrt[4]{3 x y}\)
Since the radicals are like, we combine them.
\(2 \sqrt[4]{3 x y}\)
You Try \(\PageIndex{2}\)
Simplify:
- \(4 \sqrt{3 y}-7 \sqrt{3 y}+2 \sqrt{3 y}\)
- \(6 \sqrt[3]{7 m n}+\sqrt[3]{7 m n}-4 \sqrt[3]{7 m n}\)
- Answer
-
- \(-\sqrt{3 y}\)
- \(3 \sqrt[3]{7 m n}\)
Remember that we always simplify radicals by removing the largest factor from the radicand that is a power of the index. Once each radical is simplified, we can then decide if they are like radicals.
Example \(\PageIndex{3}\)
Simplify:
- \(\sqrt{20}+3 \sqrt{5}\)
- \(\sqrt[3]{24}-\sqrt[3]{375}\)
- \(\frac{1}{2} \sqrt[4]{48}-\frac{2}{3} \sqrt[4]{243}\)
Solution :
a.
\(\sqrt{20}+3 \sqrt{5}\)
Simplify the radicals, when possible.
\(\sqrt{4} \cdot \sqrt{5}+3 \sqrt{5}\)
\(2 \sqrt{5}+3 \sqrt{5}\)
Combine the like radicals.
\(5 \sqrt{5}\)
b.
\(\sqrt[3]{24}-\sqrt[3]{375}\)
Simplify the radicals.
\(\sqrt[3]{8} \cdot \sqrt[3]{3}-\sqrt[3]{125} \cdot \sqrt[3]{3}\)
\(2 \sqrt[3]{3}-5 \sqrt[3]{3}\)
Combine the like radicals.
\(-3 \sqrt[3]{3}\)
c.
\(\frac{1}{2} \sqrt[4]{48}-\frac{2}{3} \sqrt[4]{243}\)
Simplify the radicals.
\(\frac{1}{2} \sqrt[4]{16} \cdot \sqrt[4]{3}-\frac{2}{3} \sqrt[4]{81} \cdot \sqrt[4]{3}\)
\(\frac{1}{2} \cdot 2 \cdot \sqrt[4]{3}-\frac{2}{3} \cdot 3 \cdot \sqrt[4]{3}\)
\(\sqrt[4]{3}-2 \sqrt[4]{3}\)
Combine the like radicals.
\(-\sqrt[4]{3}\)
You Try \(\PageIndex{3}\)
Simplify:
- \(\sqrt{27}+4 \sqrt{3}\)
- \(4 \sqrt[3]{5}-7 \sqrt[3]{40}\)
- \(\frac{1}{2} \sqrt[3]{128}-\frac{5}{3} \sqrt[3]{54}\)
- Answer
-
- \(7 \sqrt{3}\)
- \(-10 \sqrt[3]{5}\)
- \(-3 \sqrt[3]{2}\)
In the next example, we will remove both constant and variable factors from the radicals. Now that we have practiced taking both the even and odd roots of variables, it is common practice at this point for us to assume all variables are greater than or equal to zero so that absolute values are not needed. We will use this assumption thoughout the rest of this chapter.
Example \(\PageIndex{4}\)
Simplify:
- \(9 \sqrt{50 m^{2}}-6 \sqrt{48 m^{2}}\)
- \(\sqrt[3]{54 n^{5}}-\sqrt[3]{16 n^{5}}\)
Solution :
a.
\(9 \sqrt{50 m^{2}}-6 \sqrt{48 m^{2}}\)
Simplify the radicals.
\(9 \sqrt{25 m^{2}} \cdot \sqrt{2}-6 \sqrt{16 m^{2}} \cdot \sqrt{3}\)
\(9 \cdot 5 m \cdot \sqrt{2}-6 \cdot 4 m \cdot \sqrt{3}\)
\(45 m \sqrt{2}-24 m \sqrt{3}\)
The radicals are not like and so cannot be combined.
b.
\(\sqrt[3]{54 n^{5}}-\sqrt[3]{16 n^{5}}\)
Simplify the radicals.
\(\sqrt[3]{27 n^{3}} \cdot \sqrt[3]{2 n^{2}}-\sqrt[3]{8 n^{3}} \cdot \sqrt[3]{2 n^{2}}\)
\(3 n \sqrt[3]{2 n^{2}}-2 n \sqrt[3]{2 n^{2}}\)
Combine the like radicals.
\(n \sqrt[3]{2 n^{2}}\)
You Try \(\PageIndex{4}\)
Simplify:
- \(\sqrt{27 p^{3}}-\sqrt{48 p^{3}}\)
- \(\sqrt[3]{256 y^{5}}-\sqrt[3]{32 n^{5}}\)
- Answer
-
- \(-p \sqrt{3 p}\)
- \(4 y \sqrt[3]{4 y^{2}}-2 n \sqrt[3]{4 n^{2}}\)
Multiplying Radical Expressions
We have used the Product Property of Roots to simplify square roots by removing the perfect square factors. We can use the Product Property of Roots ‘in reverse’ to multiply square roots. Remember, we assume all variables are greater than or equal to zero.
We will rewrite the Product Property of Roots so we see both ways together.
Product Property of Roots
For any real numbers, \(\sqrt[n]{a}\) and \(\sqrt[n]{b}\), and for any integer \(n≥2\)
\(\sqrt[n]{a b}=\sqrt[n]{a} \cdot \sqrt[n]{b} \quad \text { and } \quad \sqrt[n]{a} \cdot \sqrt[n]{b}=\sqrt[n]{a b}\)
When we multiply two radicals they must have the same index. Once we multiply the radicals, we then look for factors that are a power of the index and simplify the radical whenever possible.
Multiplying radicals with coefficients is much like multiplying variables with coefficients. To multiply \(4x⋅3y\) we multiply the coefficients together and then the variables. The result is \(12xy\). Keep this in mind as you do these examples.
Example \(\PageIndex{5}\)
Simplify:
- \((6 \sqrt{2})(3 \sqrt{10})\)
- \((-5 \sqrt[3]{4})(-4 \sqrt[3]{6})\)
Solution :
a.
\((6 \sqrt{2})(3 \sqrt{10})\)
Multiply using the Product Property.
\(18\sqrt{20}\)
Simplify the radical.
\(18 \sqrt{4} \cdot \sqrt{5}\)
Simplify.
\(18 \cdot 2 \cdot \sqrt{5}\)
\(36 \sqrt{5}\)
b.
\((-5 \sqrt[3]{4})(-4 \sqrt[3]{6})\)
Multiply using the Product Property.
\(20 \sqrt[3]{24}\)
Simplify the radical.
\(20 \sqrt[3]{8} \cdot \sqrt[3]{3}\)
Simplify.
\(20 \cdot 2 \cdot \sqrt[3]{3}\)
\(40 \sqrt[3]{3}\)
You Try \(\PageIndex{5}\)
Simplify:
- \((3 \sqrt{2})(2 \sqrt{30})\)
- \((2 \sqrt[3]{18})(-3 \sqrt[3]{6})\)
- Answer
-
- \(12 \sqrt{15}\)
- \(-18 \sqrt[3]{2}\)
We follow the same procedures when there are variables in the radicands.
Example \(\PageIndex{6}\)
Simplify:
- \(\left(10 \sqrt{6 p^{3}}\right)(4 \sqrt{3 p})\)
- \(\left(2 \sqrt[4]{20 y^{2}}\right)\left(3 \sqrt[4]{28 y^{3}}\right)\)
Solution :
a.
\(\left(10 \sqrt{6 p^{3}}\right)(4 \sqrt{3 p})\)
Multiply.
\(40 \sqrt{18 p^{4}}\)
Simplify the radical.
\(40 \sqrt{9 p^{4}} \cdot \sqrt{2}\)
Simplify.
\(40 \cdot 3 p^{2} \cdot \sqrt{3}\)
\(120 p^{2} \sqrt{3}\)
b. When the radicands involve large numbers, it is often advantageous to factor them in order to find the perfect powers.
\(\left(2 \sqrt[4]{20 y^{2}}\right)\left(3 \sqrt[4]{28 y^{3}}\right)\)
Multiply.
\(6 \sqrt[4]{4 \cdot 5 \cdot 4 \cdot 7 y^{5}}\)
Simplify the radical.
\(6 \sqrt[4]{16 y^{4}} \cdot \sqrt[4]{35 y}\)
Simplify.
\(6 \cdot 2 y \sqrt[4]{35 y}\)
Multiply.
\(12 y \sqrt[4]{35 y}\)
You Try \(\PageIndex{6}\)
Simplify:
- \(\left(6 \sqrt{6 x^{2}}\right)\left(8 \sqrt{30 x^{4}}\right)\)
- \(\left(-4 \sqrt[4]{12 y^{3}}\right)\left(-\sqrt[4]{8 y^{3}}\right)\)
- Answer
-
- \(36 x^{3} \sqrt{5}\)
- \(8 y \sqrt[4]{3 y^{2}}\)
Using Polynomial Multiplication to Multiply Radical Expressions
In the next a few examples, we will use the Distributive Property to multiply expressions with radicals. First we will distribute and then simplify the radicals when possible.
Example \(\PageIndex{7}\)
Simplify:
- \(\sqrt{6}(\sqrt{2}+\sqrt{18})\)
- \(\sqrt[3]{9}(5-\sqrt[3]{18})\)
Solution :
a.
\(\sqrt{6}(\sqrt{2}+\sqrt{18})\)
Multiply.
\(\sqrt{12}+\sqrt{108}\)
Simplify.
\(\sqrt{4} \cdot \sqrt{3}+\sqrt{36} \cdot \sqrt{3}\)
Simplify.
\(2 \sqrt{3}+6 \sqrt{3}\)
Combine like radicals.
\(8\sqrt{3}\)
b.
\(\sqrt[3]{9}(5-\sqrt[3]{18})\)
Distribute.
\(5 \sqrt[3]{9}-\sqrt[3]{162}\)
Simplify.
\(5 \sqrt[3]{9}-\sqrt[3]{27} \cdot \sqrt[3]{6}\)
Simplify.
\(5 \sqrt[3]{9}-3 \sqrt[3]{6}\)
You Try \(\PageIndex{7}\)
Simplify:
- \(\sqrt{8}(2-5 \sqrt{8})\)
- \(\sqrt[3]{3}(-\sqrt[3]{9}-\sqrt[3]{6})\)
- Answer
-
- \(-40+4 \sqrt{2}\)
- \(-3-\sqrt[3]{18}\)
When we worked with polynomials, we multiplied binomials by binomials. Remember, this gave us four products before we combined any like terms. To be sure to get all four products, we organized our work—usually by the FOIL method.
Example \(\PageIndex{8}\)
Simplify:
- \((3-2 \sqrt{7})(4-2 \sqrt{7})\)
- \((\sqrt[3]{x}-2)(\sqrt[3]{x}+4)\)
Solution :
a.
\((3-2 \sqrt{7})(4-2 \sqrt{7})\)
Multiply.
\(12-6 \sqrt{7}-8 \sqrt{7}+4 \cdot 7\)
Simplify.
\(12-6 \sqrt{7}-8 \sqrt{7}+28\)
Combine like terms.
\(40-14 \sqrt{7}\)
b.
\((\sqrt[3]{x}-2)(\sqrt[3]{x}+4)\)
Multiply.
\(\sqrt[3]{x^{2}}+4 \sqrt[3]{x}-2 \sqrt[3]{x}-8\)
Combine like terms.
\(\sqrt[3]{x^{2}}+2 \sqrt[3]{x}-8\)
You Try \(\PageIndex{8}\)
Simplify:
- \((2-3 \sqrt{11})(4-\sqrt{11})\)
- \((\sqrt[3]{x}+1)(\sqrt[3]{x}+3)\)
- Answer
-
- \(41-14 \sqrt{11}\)
- \(\sqrt[3]{x^{2}}+4 \sqrt[3]{x}+3\)
Example \(\PageIndex{9}\)
Simplify: \((3 \sqrt{2}-\sqrt{5})(\sqrt{2}+4 \sqrt{5})\)
Solution :
\((3 \sqrt{2}-\sqrt{5})(\sqrt{2}+4 \sqrt{5})\)
Multiply.
\(3 \cdot 2+12 \sqrt{10}-\sqrt{10}-4 \cdot 5\)
Simplify.
\(6+12 \sqrt{10}-\sqrt{10}-20\)
Combine like terms.
\(-14+11 \sqrt{10}\)
You Try \(\PageIndex{9}\)
Simplify: \((\sqrt{6}-3 \sqrt{8})(2 \sqrt{6}+\sqrt{8})\)
- Answer
-
\(-12-20 \sqrt{3}\)
Recognizing some special products made our work easier when we multiplied binomials earlier. This is true when we multiply radicals, too. The special product formulas we used are shown here.
Special Products
Binomial Squares
\(\begin{array}{l}{(a+b)^{2}=a^{2}+2 a b+b^{2}} \\ {(a-b)^{2}=a^{2}-2 a b+b^{2}}\end{array}\)
Product of Conjugates
\((a+b)(a-b)=a^{2}-b^{2}\)
We will use the special product formulas in the next few examples. We will start with the Product of Binomial Squares Pattern .
Example \(\PageIndex{10}\)
Simplify:
- \((2+\sqrt{3})^{2}\)
- \((4-2 \sqrt{5})^{2}\)
Solution :
a.
| Multiply using the Product of Binomial Squares Pattern. | |
| Simplify. | |
| Combine like terms. |
b.
| Multiple, using the Product of Binomial Squares Pattern. | |
| Simplify. | |
| Combine like terms. |
You Try \(\PageIndex{10}\)
Simplify:
- \((10+\sqrt{2})^{2}\)
- \((9-2 \sqrt{10})^{2}\)
- Answer
-
- \(102+20 \sqrt{2}\)
- \(121-36 \sqrt{10}\)
In the next example, we will use the Product of Conjugates Pattern . Notice that the final product has no radical.
Example \(\PageIndex{11}\)
Simplify: \((5-2 \sqrt{3})(5+2 \sqrt{3})\)
Solution :
| Multiply using the Product of Conjugates Pattern. | |
| Simplify. | |
You Try \(\PageIndex{11}\)
Simplify: \((3-2 \sqrt{5})(3+2 \sqrt{5})\)
- Answer
-
\(-11\)


It was not hard to find these Fugitives; for as they fled, they were forced to fire and cut the Woods before 'em: So that Night or Day they pursu'd 'em by the Light they made, and by the Path they had cleared. But as soon as Caesar found that he was pursu'd, he put himself in a Posture of Defence, placing all the Woman and Children in the Rear; and himself, with Tuscan by his Side, or next to him, all promising to die or conquer. Encouraged thus, they never stood to parley, but fell on pell-mell upon the English , and killed some, and wounded a great many; they having Recourse to their Whips, as the best of their Weapons. And as they observed no Order, they perplexed the Enemy so sorely, with lashing 'em in the Eyes; and the Women and Children seeing their Husbands so treated, being of fearful and cowardly Dispositions, and hearing the English cry out, Yield and Live! Yield, and be Pardon'd! they all ran in amongst their Husbands and Fathers, and hung about them, crying out, Yield! Yield, and leave Caesar to their Revenge ; that by Degrees the Slaves abandon'd Caesar , and left him only Tuscan and his Heroick Imoinda , who grown as big as she was, did nevertheless press near her Lord, having a Bow and a Quiver full of poisoned Arrows, which she managed with such Dexterity, that she wounded several, and shot the Governor into the Shoulder; of which Wound he had like to have died, but that an Indian Woman, his Mistress, sucked the Wound, and cleans'd it from the Venom: But however, he stir'd not from the Place till he had parly'd with Caesar , who he found was resolved to die fighting, and would not be taken; no more would Tuscan or Imoinda . But he, more thirsting after Revenge of another Sort, than that of depriving him of Life, now made use of all his Art of Talking and Dissembling, and besought Caesar to yield himself upon Terms which he himself should propose, and should be sacredly assented to, and kept by him. He told him, It was not that he any longer fear'd him, or could believe the Force of two Men, and a young Heroine, could overthrow all them, and with all the Slaves now on their Side also; but it was the vast Esteem he had for his Person, the Desire he had to serve so gallant a Man, and to hinder himself from the Reproach hereafter, of having been the Occasion of the Death of a Prince, whose Valour and Magnanimity deserved the Empire of the World. He protested to him, he looked upon his Action as gallant and brave, however tending to the Prejudice of his Lord and Master, who would by it have lost so considerable a Number of Slaves; that this Flight of his should be look'd on as a Heat of Youth, and a Rashness of a too forward Courage, and an unconsider'd Impatience of Liberty, and no more; and that he labour'd in vain to accomplish that which they would effectually perform as soon as any Ship arrived that would touch on his Coast: 'So that if you will be pleased ( continued he ) to surrender yourself, all imaginable Respect shall be paid you; and your Self, your Wife and Child, if it be born here, shall depart free out of our Land.' But Caesar would hear of no Composition; though Byam urged, if he pursued and went on in his Design, he would inevitably perish, either by great Snakes, wild Beasts or Hunger; and he ought to have Regard to his Wife, whose Condition requir'd Ease, and not the Fatigues of tedious Travel, where she could not be secured from being devoured. But Caesar told him, there was no Faith in the White men, or the Gods they ador'd; who instructed them in Principles so false, that honest Men could not live amongst them; though no People profess'd so much, none perform'd so little: That he knew what he had to do when he dealt with Men of Honour; but with them a Man ought to be eternally on his Guard, and never to eat and drink with Christians, without his Weapon of Defence in his Hand; and, for his own Security, never to credit one Word they spoke. As for the Rashness and Inconsiderateness of his Action, he would confess the Governor is in the right; and that he was ashamed of what he had done in endeavouring to make those free, who were by Nature Slaves, poor wretched Rogues, fit to be used as Christian Tools; Dogs, treacherous and cowardly, fit for such Masters; and they wanted only but to be whipped into the Knowledge of the Christian Gods, to be the vilest of all creeping Things; to learn to worship such Deities as had not Power to make them just, brave, or honest: In fine, after a thousand Things of this Nature, not fit here to be recited, he told Byam , He had rather die, than live upon the same Earth with such Dogs. But Trefry and Byam pleaded and protested together so much, that Trefry believing the Governor to mean what he said, and speaking very cordially himself, generously put himself into Caesar's Hands, and took him aside, and persuaded him, even with Tears, to live, by surrendring himself, and to name his Conditions. Caesar was overcome by his Wit and Reasons, and in Consideration of Imoinda ; and demanding what he desired, and that it should be ratify'd by their Hands in Writing, because he had perceived that was the common Way of Contract between Man and Man amongst the Whites; all this was performed, and Tuscan's Pardon was put in, and they surrender'd to the Governor, who walked peaceably down into the Plantation with them, after giving Order to bury their Dead. Caesar was very much toil'd with the Bustle of the Day, for he had fought like a Fury; and what Mischief was done, he and Tuscan performed alone; and gave their Enemies a fatal Proof, that they durst do any Thing, and fear'd no mortal Force.
But they were no sooner arrived at the Place where all the Slaves receive their Punishments of Whipping, but they laid Hands on Caesar and Tuscan , faint with Heat and Toil; and surprizing them, bound them to two several Stakes, and whipped them in a most deplorable and inhuman Manner, rending the very Flesh from their Bones, especially Caesar , who was not perceived to make any Moan, or to alter his Face, only to roll his Eyes on the faithless Governor, and those he believed Guilty, with Fierceness and Indignation; and to complete his Rage, he saw every one of those Slaves who but a few Days before ador'd him as something more than Mortal, now had a Whip to give him some Lashes, while he strove not to break his Fetters; tho' if he had, it were impossible: but he pronounced a Woe and Revenge from his Eyes, that darted Fire, which was at once both aweful and terrible to behold.
When they thought they were sufficiently revenged on him, they unty'd him, almost fainting with Loss of Blood, from a thousand Wounds all over his Body; from which they had rent his Clothes, and led him bleeding and naked as he was, and loaded him all over with Irons; and then rubb'd his Wounds, to complete their Cruelty, with Indian Pepper, which had like to have made him raving mad; and, in this Condition made him so fast to the Ground, that he could not stir, if his Pains and Wounds would have given him Leave. They spared Imoinda , and did not let her see this Barbarity committed towards her Lord, but carried her down to Parham , and shut her up; which was not in Kindness to her, but for Fear she should die with the Sight, or miscarry, and then they should lose a young Slave, and perhaps the Mother.
You must know, that when the News was brought on Monday Morning, that Caesar had betaken himself to the Woods, and carry'd with him all the Negroes , we were possess'd with extreme Fear, which no Persuasions could dissipate, that he would secure himself till Night, and then would come down and cut all our Throats. This Apprehension made all the Females of us fly down the River, to be secured; and while we were away, they acted this Cruelty; for I suppose I had Authority and Interest enough there, had I suspected any such Thing, to have prevented it: but we had not gone many Leagues, but the News overtook us, that Caesar was taken and whipped liked a common Slave. We met on the River with Colonel Martin , a Man of great Gallantry, Wit, and Goodness, and whom I have celebrated in a Character of my new Comedy, by his own Name, in Memory of so brave a Man: He was wise and eloquent, and, from the Fineness of his Parts, bore a great Sway over the Hearts of all the Colony: He was a Friend to Caesar , and resented this false Dealing with him very much. We carried him back to Parham , thinking to have made an Accommodation; when he came, the first News we heard, was, That the Governor was dead of a Wound Imoinda had given him; but it was not so well. But it seems, he would have the Pleasure of beholding the Revenge he took on Caesar ; and before the cruel Ceremony was finished, he dropt down; and then they perceived the Wound he had on his Shoulder was by a venom'd Arrow, which, as I said, his Indian Mistress healed by sucking the Wound.
We were no sooner arrived, but we went up to the Plantation to see Caesar ; whom we found in a very miserable and unexpressible Condition; and I have a thousand Times admired how he lived in so much tormenting Pain. We said all Things to him, that Trouble, Pity and Good-Nature could suggest, protesting our Innocency of the Fact, and our Abhorrence of such Cruelties; making a thousand Professions and Services to him, and begging as many Pardons for the Offenders, till we said so much, that he believed we had no Hand in his ill Treatment; but told us, He could never pardon Byam ; as for Trefry , he confess'd he saw his Grief and Sorrow for his Suffering, which he could not hinder, but was like to have been beaten down by the very Slaves, for speaking in his Defence: But for Byam , who was their Leader, their Head—and should, by his Justice and Honour, have been an Example to 'em—for him, he wished to live to take a dire Revenge of him; and said, It had been well for him, if he had sacrificed me, instead of giving me the contemptible Whip. He refused to talk much; but begging us to give him our Hands, he took them, and protested never to lift up his to do us any Harm. He had a great Respect for Colonel Martin , and always took his Counsel like that of a Parent; and assured him, he would obey him in any Thing but his Revenge on Byam : 'Therefore ( said he ) for his own Safety, let him speedily dispatch me; for if I could dispatch myself, I would not, till that Justice were done to my injured Person, and the Contempt of a Soldier: No, I would not kill myself, even after a Whipping, but will be content to live with that Infamy, and be pointed at by every grinning Slave, till I have completed my Revenge; and then you shall see, that Oroonoko scorns to live with the Indignity that was put on Caesar .' All we could do, could get no more Words from him; and we took Care to have him put immediately into a healing Bath, to rid him of his Pepper, and ordered a Chirurgeon to anoint him with healing Balm, which he suffer'd, and in some Time he began to be able to walk and eat. We failed not to visit him every Day, and to that End had him brought to an Apartment at Parham .
The Governor had no sooner recover'd, and had heard of the Menaces of Caesar , but he called his Council, who (not to disgrace them, or burlesque the Government there) consisted of such notorious Villains as Newgate never transported; and, possibly, originally were such who understood neither the Laws of God or Man, and had no sort of Principles to make them worthy the Name of Men; but at the very Council-Table would contradict and fight with one another, and swear so bloodily, that 'twas terrible to hear and see 'em. (Some of 'em were afterwards hanged, when the Dutch took Possession of the Place, others sent off in Chains.) But calling these special Rulers of the Nation together, and requiring their Counsel in this weighty Affair, they all concluded, that (damn 'em) it might be their own Cases; and that Caesar ought to be made an Example to all the Negroes , to fright 'em from daring to threaten their Betters, their Lords and Masters; and at this Rate no Man was safe from his own Slaves; and concluded, nemine contradicente , That Caesar should be hanged.
Trefry then thought it Time to use his Authority, and told Byam , his Command did not extend to his Lord's Plantation; and that Parham was as much exempt from the Law as White-Hall ; and that they ought no more to touch the Servants of the Lord—(who there represented the King's Person) than they could those about the King himself; and that Parham was a Sanctuary; and tho' his Lord were absent in Person, his Power was still in being there, which he had entrusted with him, as far as the Dominions of his particular Plantations reached, and all that belonged to it; the rest of the Country, as Byam was Lieutenant to his Lord, he might exercise his Tyranny upon. Trefry had others as powerful, or more, that interested themselves in Caesar's Life, and absolutely said, he should be defended. So turning the Governor, and his wise Council, out of Doors, (for they sat at Parham-House ) we set a Guard upon our Lodging-Place, and would admit none but those we called Friends to us and Caesar .
The Governor having remain'd wounded at Parham , till his Recovery was completed, Caesar did not know but he was still there, and indeed for the most Part, his Time was spent there: for he was one that loved to live at other Peoples Expence, and if he were a Day absent, he was ten present there; and us'd to play, and walk, and hunt, and fish with Caesar : So that Caesar did not at all doubt, if he once recover'd Strength, but he should find an Opportunity of being revenged on him; though, after such a Revenge, he could not hope to live: for if he escaped the Fury of the English Mobile, who perhaps would have been glad of the Occasion to have killed him, he was resolved not to survive his Whipping; yet he had some tender Hours, a repenting Softness, which he called his Fits of Cowardice, wherein he struggled with Love for the Victory of his Heart, which took Part with his charming Imoinda there; but for the most Part, his Time was pass'd in melancholy Thoughts, and black Designs. He consider'd, if he should do this Deed, and die either in the Attempt, or after it, he left his lovely Imoinda a Prey, or at best a Slave to the enraged Multitude; his great Heart could not endure that Thought: Perhaps (said he) she may be first ravish'd by every Brute; expos'd first to their nasty Lusts, and then a shameful Death : No, he could not live a Moment under that Apprehension, too insupportable to be borne. These were his Thoughts, and his silent Arguments with his Heart, as he told us afterwards: So that now resolving not only to kill Byam , but all those he thought had enraged him; pleasing his great Heart with the fancy'd Slaughter he should make over the whole Face of the Plantation; he first resolved on a Deed, (that however horrid it first appear'd to us all) when we had heard his Reasons, we thought it brave and just. Being able to walk, and, as he believed, fit for the Execution of his great Design, he begg'd Trefry to trust him into the Air, believing a Walk would do him good; which was granted him; and taking Imoinda with him, as he used to do in his more happy and calmer Days, he led her up into a Wood, where (after with a thousand Sighs, and long gazing silently on her Face, while Tears gush'd, in spite of him, from his Eyes) he told her his Design, first of killing her, and then his Enemies, and next himself, and the Impossibility of escaping, and therefore he told her the Necessity of dying. He found the heroick Wife faster pleading for Death, than he was to propose it, when she found his fix'd Resolution; and, on her Knees, besought him not to leave her a Prey to his Enemies. He (grieved to Death) yet pleased at her noble Resolution, took her up, and embracing of her with all the Passion and Languishment of a dying Lover, drew his Knife to kill this Treasure of his Soul, this Pleasure of his Eyes; while Tears trickled down his Cheeks, hers were smiling with Joy she should die by so noble a Hand, and be sent into her own Country (for that's their Notion of the next World) by him she so tenderly loved, and so truly ador'd in this: For Wives have a Respect for their Husbands equal to what any other People pay a Deity; and when a Man finds any Occasion to quit his Wife, if he love her, she dies by his Hand; if not, he sells her, or suffers some other to kill her. It being thus, you may believe the Deed was soon resolv'd on; and 'tis not to be doubted, but the parting, the eternal Leave-taking of two such Lovers, so greatly born, so sensible, so beautiful, so young, and so fond, must be very moving, as the Relation of it was to me afterwards.
All that Love could say in such Cases, being ended, and all the intermitting Irresolutions being adjusted, the lovely, young and ador'd Victim lays herself down before the Sacrificer; while he, with a Hand resolved, and a Heart-breaking within, gave the fatal Stroke, first cutting her Throat, and then severing her yet smiling Face from that delicate Body, pregnant as it was with the Fruits of tenderest Love. As soon as he had done, he laid the Body decently on Leaves and Flowers, of which he made a Bed, and conceal'd it under the same Cover-lid of Nature; only her Face he left yet bare to look on: But when he found she was dead, and past all Retrieve, never more to bless him with her Eyes, and soft Language, his Grief swell'd up to Rage; he tore, he rav'd, he roar'd like some Monster of the Wood, calling on the lov'd Name of Imoinda . A thousand Times he turned the fatal Knife that did the Deed towards his own Heart, with a Resolution to go immediately after her; but dire Revenge, which was now a thousand Times more fierce in his Soul than before, prevents him; and he would cry out, 'No, since I have sacrific'd Imoinda to my Revenge, shall I lose that Glory which I have purchased so dear, as at the Price of the fairest, dearest, softest Creature that ever Nature made? No, no!' Then at her Name Grief would get the Ascendant of Rage, and he would lie down by her Side, and water her Face with Showers of Tears, which never were wont to fall from those Eyes; and however bent he was on his intended Slaughter, he had not Power to stir from the Sight of this dear Object, now more beloved, and more ador'd than ever.
He remained in this deplorable Condition for two Days, and never rose from the Ground where he had made her sad Sacrifice; at last rouzing from her Side, and accusing himself with living too long, now Imoinda was dead, and that the Deaths of those barbarous Enemies were deferred too long, he resolved now to finish the great Work: but offering to rise, he found his Strength so decay'd, that he reeled to and fro, like Boughs assailed by contrary Winds; so that he was forced to lie down again, and try to summon all his Courage to his Aid. He found his Brains turned round, and his Eyes were dizzy, and Objects appear'd not the same to him they were wont to do; his Breath was short, and all his Limbs surpriz'd with a Faintness he had never felt before. He had not eat in two Days, which was one Occasion of his Feebleness, but Excess of Grief was the greatest; yet still he hoped he should recover Vigour to act his Design, and lay expecting it yet six Days longer; still mourning over the dead Idol of his Heart, and striving every Day to rise, but could not.
In all this time you may believe we were in no little Affliction for Caesar and his Wife; some were of Opinion he was escaped, never to return; others thought some Accident had happened to him: But however, we fail'd not to send out a hundred People several Ways, to search for him. A Party of about forty went that Way he took, among whom was Tuscan , who was perfectly reconciled to Byam : They had not gone very far into the Wood, but they smelt an unusual Smell, as of a dead Body; for Stinks must be very noisom, that can be distinguish'd among such a Quantity of natural Sweets, as every Inch of that Land produces: so that they concluded they should find him dead, or some body that was so; they pass'd on towards it, as loathsom as it was, and made such rustling among the Leaves that lie thick on the Ground, by continual falling, that Caesar heard he was approach'd; and though he had, during the Space of these eight Days, endeavour'd to rise, but found he wanted Strength, yet looking up, and seeing his Pursuers, he rose, and reel'd to a neighbouring Tree, against which he fix'd his Back; and being within a dozen Yards of those that advanc'd and saw him, he call'd out to them, and bid them approach no nearer, if they would be safe. So that they stood still, and hardly believing their Eyes, that would persuade them that it was Caesar that spoke to them, so much he was alter'd; they ask'd him, what he had done with his Wife, for they smelt a Stink that almost struck them dead? He pointing to the dead Body, sighing, cry'd, Behold her there. They put off the Flowers that cover'd her, with their Sticks, and found she was kill'd, and cry'd out, Oh, Monster! that hast murder'd thy Wife. Then asking him, why he did so cruel a Deed? He reply'd, He had no Leisure to answer impertinent Questions: 'You may go back ( continued he ) and tell the faithless Governor, he may thank Fortune that I am breathing my last; and that my Arm is too feeble to obey my Heart, in what it had design'd him': But his Tongue faultering, and trembling, he could scarce end what he was saying. The English taking Advantage by his Weakness, cry'd, Let us take him alive by all Means. He heard 'em; and, as if he had reviv'd from a Fainting, or a Dream, he cried out, 'No, Gentlemen, you are deceived; you will find no more Caesars to be whipt; no more find a Faith in me; Feeble as you think me, I have Strength yet left to secure me from a second Indignity.' They swore all anew; and he only shook his Head, and beheld them with Scorn. Then they cry'd out, Who will venture on this single Man? Will nobody? They stood all silent, while Caesar replied, Fatal will be the Attempt of the first Adventurer, let him assure himself , (and, at that Word, held up his Knife in a menacing Posture:) Look ye, ye faithless Crew , said he, 'tis not Life I seek, nor am I afraid of dying , (and at that Word, cut a Piece of Flesh from his own Throat, and threw it at 'em) yet still I would live if I could, till I had perfected my Revenge: But, oh! it cannot be; I feel Life gliding from my Eyes and Heart; and if I make not haste, I shall fall a Victim to the shameful Whip.  But when they saw him tottering, they cry'd out, Will none venture on him? A bold Englishman cry'd, Yes, if he were the Devil , (taking Courage when he saw him almost dead) and swearing a horrid Oath for his farewel to the World, he rush'd on him. Caesar with his arm'd Hand, met him so fairly, as stuck him to the Heart, and he Fell dead at his feet. Tuscan seeing that, cry'd out, I love thee, O Caesar ! and therefore will not let thee die, if possible ; and running to him, took him in his Arms; but, at the same time, warding a Blow that Caesar made at his Bosom, he receiv'd it quite through his Arm; and Caesar having not Strength to pluck the Knife forth, tho' he attempted it, Tuscan neither pull'd it out himself, nor suffer'd it to be pull'd out, but came down with it sticking in his Arm; and the Reason he gave for it, was, because the Air should not get into the Wound. They put their Hands a-cross, and carry'd Caesar between six of 'em, fainting as he was, and they thought dead, or just dying; and they brought him to Parham , and laid him on a Couch, and had the Chirurgeon immediately to him, who dressed his Wounds, and sow'd up his Belly, and us'd Means to bring him to Life, which they effected. We ran all to see him; and, if before we thought him so beautiful a Sight, he was now so alter'd, that his Face was like a Death's-Head black'd over, nothing but Teeth and Eye-holes: For some Days we suffer'd no Body to speak to him, but caused Cordials to be poured down his Throat; which sustained his Life, and in six or seven Days he recovered his Senses: For, you must know, that Wounds are almost to a Miracle cur'd in the Indies ; unless Wounds in the Legs, which they rarely ever cure.
When he was well enough to speak, we talk'd to him, and ask'd him some Questions about his Wife, and the Reasons why he kill'd her; and he then told us what I have related of that Resolution, and of his Parting, and he besought us we would let him die, and was extremely afflicted to think it was possible he might live: He assur'd us, if we did not dispatch him, he would prove very fatal to a great many. We said all we could to make him live, and gave him new Assurances; but he begg'd we would not think so poorly of him, or of his Love to Imoinda , to imagine we could flatter him to Life again: But the Chirurgeon assur'd him he could not live, and therefore he need not fear. We were all (but Caesar ) afflicted at this News, and the Sight was ghastly: His Discourse was sad; and the earthy Smell about him so strong, that I was persuaded to leave the Place for some time, (being my self but sickly, and very apt to fall into Fits of dangerous Illness upon any extraordinary Melancholy.) The Servants, and Trefry , and the Chirurgeons, promis'd all to take what possible Care they could of the Life of Caesar ; and I, taking Boat, went with other Company to Colonel Martin's , about three Days Journey down the River. But I was no sooner gone, than the Governor taking Trefry , about some pretended earnest Business, a Day's Journey up the River, having communicated his Design to one Banister , a wild Irish Man, one of the Council, a Fellow of absolute Barbarity, and fit to execute any Villany, but rich; he came up to Parham , and forcibly took Caesar , and had him carried to the same Post where he was whipp'd; and causing him to be ty'd to it, and a great Fire made before him, he told him he should die like a Dog, as he was. Caesar replied, This was the first piece of Bravery that ever Banister did, and he never spoke Sense till he pronounc'd that Word; and if he would keep it, he would declare, in the other World, that he was the only Man, of all the Whites , that ever he heard speak Truth. And turning to the Men that had bound him, he said, My Friends, am I to die, or to be whipt? And they cry'd, Whipt! no, you shall not escape so well. And then he reply'd, smiling, A Blessing on thee ; and assur'd them they need not tie him, for he would stand fix'd like a Rock, and endure Death so as should encourage them to die: But if you whip me (said he) be sure you tie me fast .
 My Mother and Sister were by him all the While, but not suffer'd to save him; so rude and wild were the Rabble, and so inhuman were the Justices who stood by to see the Execution, who after paid dear enough for their Insolence. They cut Caesar into Quarters, and sent them to several of the chief Plantations: One Quarter was sent to Colonel Martin ; who refus'd it, and swore, he had rather see the Quarters of Banister , and the Governor himself, than those of Caesar , on his Plantations; and that he could govern his Negroes , without terrifying and grieving them with frightful Spectacles of a mangled King.
Thus died this great Man, worthy of a better Fate, and a more sublime Wit than mine to write his Praise: Yet, I hope, the Reputation of my Pen is considerable enough to make his glorious Name to survive to all Ages, with that of the brave, the beautiful and the constant Imoinda .
Notes: Critical and Explanatory Epistle Dedicatory Richard Maitland, fourth Earl of Lauderdale (1653-95), eldest son of Charles, third Earl of Lauderdale by Elizabeth, daughter and heiress of Richard Lauder of Halton, was born 20 June, 1653. Before his father succeeded to the Lauderdale title he was styled of Over-Gogar; after that event he was known as Lord Maitland. 9 October, 1678, he was sworn a Privy Councillor, and appointed Joint General of the Mint with his father. In 1681 he was made Lord Justice General, but deprived of that office three years later on account of suspected communications with his father-in-law, Argyll, who had fled to Holland in 1681. Maitland, however, was in truth a strong Jacobite, and refusing to accept the Revolution settlement became an exile with his King. He is said to have been present at the battle of the Boyne, 1 July, 1690. He resided for some time at St. Germains, but fell into disfavour, perhaps owing to the well-known protestant sympathies of his wife, Lady Agnes Campbell (1658-1734), second daughter of the fanatical Archibald, Earl of Argyll. From St. Germains Maitland retired to Paris, where he died in 1695. He had succeeded to the Earldom of Lauderdale 9 June, 1691, but was outlawed by the Court of Justiciary, 23 July, 1694. He left no issue. Lauderdale was the author of a verse translation of Virgil (8vo, 1718 and 2 Vols., 12mo, 1737). Dryden, to whom he sent a MS. copy from Paris, states that whilst working on his own version he consulted this whenever a crux appeared in the Latin text. Lauderdale also wrote A Memorial on the Estate of Scotland (about 1690), printed in Hooke's Correspondence (Roxburghe Club), and there wrongly ascribed to the third Earl, his father. [back] I gave 'em to the King's Theatre Sir Robert Howard and Dryden's heroic tragedy, The Indian Queen , was produced at the Theatre Royal in mid-January, 1663. It is a good play, but the extraordinary success it attained was in no small measure due to the excellence and magnificence of the scenic effects and mounting. 27 January, Pepys noticed that the streets adjacent to the theatre were 'full of coaches at the new play The Indian Queen , which for show, they say, exceeds Henry VIII .' On 1 February he himself found it 'indeed a most pleasant show'. The grandeur of the mise en scène became long proverbial in theatrical history. Zempoalla, the Indian Queen, a fine rôle, was superbly acted by Mrs. Marshall, the leading tragedienne of the day. The feathered ornaments which Mrs. Behn mentions must have formed a quaint but doubtless striking addition to the actress's pseudo-classic attire. Bernbaum pictures 'Nell Gwynn in the true costume of a Carib belle', a quite unfair deduction from Mrs. Behn's words. [back] Osenbrigs More usually 'osnaburg', so named from Osnabrück in North Germany, a kind of coarse linen made in this town. Narborough's Journal, 1669 ( An Account of Several Late Voyages , 1694), speaks of 'Cloth, Osenbrigs, Tobacco'. cf. Pennsylvania Col. Records (1732): 'That to each there be given a couple of Shirts, a Jackett, two pairs of trowsers of Oznabrigs.' [back] as soon as the Governour arrived The Governor was Francis Willoughby, fifth Baron Willoughby of Parham (1613?-1666). He had arrived at Barbadoes, 29 April, 1650, and was received as Governor 7 May, which same day he caused Charles II to be proclaimed. An ardent royalist, he was dispossessed by an Act of Parliament, 4 March, 1652, and summoned back to England. At the Restoration he was reinstated, and arrived the second time with full powers in Barbadoes, 10 August, 1663. About the end of July, 1666, he was lost at sea on board the good ship Hope . [back] my Father... never arriv'd to possess the Honour design'd him Bernbaum, following the mistaken statement that Mrs. Behn's father, John Amis, was a barber, argues that a man in such a position could hardly have obtained so important a post, and if her 'father was not sent to Surinam, the only reason she gives for being there disappears.' However, since we know her father to have been no barber, but of good family, this line of discussion falls to the ground. [back] Brother to Harry Martin the great Oliverian Henry, or Harry, and George Marten were the two sons of Sir Henry Marten ( ob. 1641) and his first wife, Elizabeth, who died 19 June, 1618. [back] The Deputy-Governor William Byam was 'Lieutenant General of Guiana and Governor of Willoughby Land', 1661-7. Even previously to this he had gained no little influence and power in these colonies. He headed the forces that defended Surinam in 1667 against the Dutch Admiral Crynsens, who, however, proved victorious. [back] my new Comedy The Younger Brother; or, The Amorous Jilt , posthumously produced under the auspices of, and with some alterations by, Charles Gildon at Drury Lane in 1696. George Marteen, acted by Powell, is the young and gallant hero of the comedy. [back] his Council In The Widow Ranter Mrs. Behn draws a vivid picture of these deboshed ruffians. [back] one Banister Sergeant Major James Banister being, after Byam's departure in 1667, 'the only remaining eminent person' became Lieutenant-Governor. It was he who in 1668 made the final surrender of the colony. Later, having quarrelled with the Dutch he was imprisoned by them. [back]


<IP_ADDRESS>: Charles Darwin and Natural Selection
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Learning Objectives
- Explain how natural selection can lead to evolution
Charles Darwin and Natural Selection
In the mid-nineteenth century, the mechanism for evolution was independently conceived of and described by two naturalists: Charles Darwin and Alfred Russel Wallace. Importantly, each naturalist spent time exploring the natural world on expeditions to the tropics. From 1831 to 1836, Darwin traveled around the world to places like South America, Australia, and the southern tip of Africa. Wallace traveled to Brazil to collect insects in the Amazon rainforest from 1848 to 1852 and to the Malay Archipelago from 1854 to 1862. Darwin’s journey, as with Wallace’s later journeys to the Malay Archipelago, included stops at several island chains, the last being the Galápagos Islands west of Ecuador. On these islands, Darwin observed that species of organisms on different islands were clearly similar, yet had distinct differences. For example, the ground finches inhabiting the Galápagos Islands comprised several species with a unique beak shape. The species on the islands had a graded series of beak sizes and shapes with very small differences between the most similar. He observed that these finches closely resembled another finch species on the mainland of South America. Darwin imagined that the island species might be modified from one of the original mainland species. Upon further study, he realized that the varied beaks of each finch helped the birds acquire a specific type of food. For example, seed-eating finches had stronger, thicker beaks for breaking seeds, while insect-eating finches had spear-like beaks for stabbing their prey.
Natural Selection
Wallace and Darwin observed similar patterns in other organisms and independently developed the same explanation for how and why such changes could take place. Darwin called this mechanism natural selection. Natural selection, also known as “survival of the fittest,” is the more prolific reproduction of individuals with favorable traits that survive environmental change because of those traits. This leads to evolutionary change, the trait becoming predominant within a population. For example, Darwin observed that a population of giant tortoises found in the Galapagos Archipelago have longer necks than those that lived on other islands with dry lowlands. These tortoises were “selected” because they could reach more leaves and access more food than those with short necks. In times of drought, when fewer leaves would be available, those that could reach more leaves had a better chance to eat and survive than those that could not reach the food source. Consequently, long-necked tortoises would more probably be reproductively successful and pass the long-necked trait to their offspring. Over time, only long-necked tortoises would be present in the population.
Natural selection, Darwin argued, was an inevitable outcome of three principles that operated in nature. First, most characteristics of organisms are inherited, or passed from parent to offspring, although how traits were inherited was unknown. Second, more offspring are produced than are able to survive. The capacity for reproduction in all organisms outstrips the availability of resources to support their numbers. Thus, there is competition for those resources in each generation. Both Darwin and Wallace were influenced by an essay written by economist Thomas Malthus who discussed this principle in relation to human populations. Third, Darwin and Wallace reasoned that offspring with the inherited characteristics that allow them to best compete for limited resources will survive and have more offspring than those individuals with variations that are less able to compete. Because characteristics are inherited, these traits will be better represented in the next generation. This will lead to change in populations over successive generations in a process that Darwin called descent with modification. Ultimately, natural selection leads to greater adaptation of the population to its local environment; it is the only mechanism known for adaptive evolution.
Papers by Darwin and Wallace presenting the idea of natural selection were read together in 1858 before the Linnean Society in London. The following year, Darwin’s book, On the Origin of Species , was published. His book outlined his arguments for evolution by natural selection.
Key Points
- Wallace traveled to Brazil to collect and observe insects from the Amazon rainforest.
- Darwin observed that finches in the Galápagos Islands had different beaks than finches in South America; these adaptations equiped the birds to acquire specific food sources.
- Wallace and Darwin observed similar patterns in the variation of organisms and independently developed the same explanation for how such variations could occur over time, a mechanism Darwin called natural selection.
- According to natural selection, also known as “survival of the fittest,” individuals with traits that enable them to survive are more reproductively successful; this leads to those traits becoming predominant within a population.
- Natural selection is an inevitable outcome of three principles: most characteristics are inherited, more offspring are produced than are able to survive, and offspring with more favorable characteristics will survive and have more offspring than those individuals with less favorable traits.
Key Terms
- natural selection : a process in which individual organisms or phenotypes that possess favorable traits are more likely to survive and reproduce
- descent with modification : change in populations over generations


8.8: Magnetic Force on a Current-Carrying Conductor
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Learning Objectives
By the end of this section, you will be able to:
- Determine the direction in which a current-carrying wire experiences a force in an external magnetic field
- Calculate the force on a current-carrying wire in an external magnetic field
Moving charges experience a force in a magnetic field. If these moving charges are in a wire—that is, if the wire is carrying a current—the wire should also experience a force. However, before we discuss the force exerted on a current by a magnetic field, we first examine the magnetic field generated by an electric current. We are studying two separate effects here that interact closely: A current-carrying wire generates a magnetic field and the magnetic field exerts a force on the current-carrying wire.
Magnetic Fields Produced by Electrical Currents
When discussing historical discoveries in magnetism, we mentioned Oersted’s finding that a wire carrying an electrical current caused a nearby compass to deflect. A connection was established that electrical currents produce magnetic fields. (This connection between electricity and magnetism is discussed in more detail in Sources of Magnetic Fields .)
The compass needle near the wire experiences a force that aligns the needle tangent to a circle around the wire. Therefore, a current-carrying wire produces circular loops of magnetic field. To determine the direction of the magnetic field generated from a wire, we use a second right-hand rule. In RHR-2, your thumb points in the direction of the current while your fingers wrap around the wire, pointing in the direction of the magnetic field produced (Figure \(\PageIndex{1}\)). If the magnetic field were coming at you or out of the page, we represent this with a dot. If the magnetic field were going into the page, we represent this with an × .
These symbols come from considering a vector arrow: An arrow pointed toward you, from your perspective, would look like a dot or the tip of an arrow. An arrow pointed away from you, from your perspective, would look like a cross or an × . A composite sketch of the magnetic circles is shown in Figure \(\PageIndex{1}\), where the field strength is shown to decrease as you get farther from the wire by loops that are farther separated.
Calculating the Magnetic Force
Electric current is an ordered movement of charge. A current-carrying wire in a magnetic field must therefore experience a force due to the field. To investigate this force, let’s consider the infinitesimal section of wire as shown in Figure \(\PageIndex{3}\). The length and cross-sectional area of the section are dl and A , respectively, so its volume is \(V = A \cdot dl\). The wire is formed from material that contains n charge carriers per unit volume, so the number of charge carriers in the section is \(nA \cdot dl\). If the charge carriers move with drift velocity \(\vec{v}_d\) the current I in the wire is (from Current and Resistance )
\[I = neAv_d. \nonumber \]
The magnetic force on any single charge carrier is \(e\vec{v}_d \times \vec{B}\), so the total magnetic force \(d\vec{F}\) on the \(nA\cdot dl\) charge carriers in the section of wire is
\[d\vec{F} = (nA \cdot dl)e\vec{v}_d \times \vec{B}. \nonumber \]
We can define dl to be a vector of length dl pointing along \(\vec{v}_d\), which allows us to rewrite this equation as
\[d\vec{F} = neAv_dd\vec{l} \times \vec{B}, \nonumber \] or
\[d\vec{F} = Id\vec{l} \times \vec{B}. \label{11.12} \]
This is the magnetic force on the section of wire. Note that it is actually the net force exerted by the field on the charge carriers themselves. The direction of this force is given by RHR-1, where you point your fingers in the direction of the current and curl them toward the field. Your thumb then points in the direction of the force.
To determine the magnetic force \(\vec{F}\) on a wire of arbitrary length and shape, we must integrate Equation \ref{11.12} over the entire wire. If the wire section happens to be straight and B is uniform, the equation differentials become absolute quantities, giving us
\[\vec{F} = I\vec{l} \times \vec{B}. \nonumber \]
This is the force on a straight, current-carrying wire in a uniform magnetic field.
Example \(\PageIndex{1}\): Balancing the Gravitational and Magnetic Forces on a Current-Carrying Wire
A wire of length 50 cm and mass 10 g is suspended in a horizontal plane by a pair of flexible leads (Figure \(\PageIndex{3}\)). The wire is then subjected to a constant magnetic field of magnitude 0.50 T, which is directed as shown. What are the magnitude and direction of the current in the wire needed to remove the tension in the supporting leads?
Strategy
From the free-body diagram in the figure, the tensions in the supporting leads go to zero when the gravitational and magnetic forces balance each other. Using the RHR-1, we find that the magnetic force points up. We can then determine the current I by equating the two forces.
Solution
Equate the two forces of weight and magnetic force on the wire:
\[mg = IlB. \nonumber \] Thus,
\[I = \frac{mg}{lB} = \frac{(0.010 \, kg}{9.8 \, m/s^2)}{(0.50 \, m)(0.50 \, T)} = 0.39 \, A. \nonumber \]
Significance
This large magnetic field creates a significant force on a length of wire to counteract the weight of the wire.
Example \(\PageIndex{2}\): Calculating Magnetic Force on a Current-Carrying Wire
A long, rigid wire lying along the y -axis carries a 5.0-A current flowing in the positive y -direction. (a) If a constant magnetic field of magnitude 0.30 T is directed along the positive x -axis, what is the magnetic force per unit length on the wire? (b) If a constant magnetic field of 0.30 T is directed 30 degrees from the + x -axis towards the + y -axis, what is the magnetic force per unit length on the wire?
Strategy
The magnetic force on a current-carrying wire in a magnetic field is given by \(\vec{F} = I\vec{l} \times \vec{B}\). For part a, since the current and magnetic field are perpendicular in this problem, we can simplify the formula to give us the magnitude and find the direction through the RHR-1. The angle θ is 90 degrees, which means \(sin \, \theta = 1.\) Also, the length can be divided over to the left-hand side to find the force per unit length. For part b, the current times length is written in unit vector notation, as well as the magnetic field. After the cross product is taken, the directionality is evident by the resulting unit vector.
Solution
- We start with the general formula for the magnetic force on a wire. We are looking for the force per unit length, so we divide by the length to bring it to the left-hand side. We also set \(sin \, \theta\). The solution therefore is \[F = IlB \, sin \, \theta \nonumber \]\[\frac{F}{l} = (5.0 \, A)(0.30 \, T) \nonumber \] \[\frac{F}{l} = 1.5 \, N/m. \nonumber \] Directionality: Point your fingers in the positive y -direction and curl your fingers in the positive x -direction. Your thumb will point in the \(-\vec{k}\) direction. Therefore, with directionality, the solution is \[\frac{\vec{F}}{l} = -1.5 \vec{k} \, N/m. \nonumber \]
- The current times length and the magnetic field are written in unit vector notation. Then, we take the cross product to find the force: \[\vec{F} = I\vec{l} \times \vec{B} = (5.0 A) l\hat{j} \times (0.30 T \, cos(30^o)\hat{i} \nonumber \] \[\vec{F}/l = -1.30 \hat{k} \, N/m. \nonumber \]
Significance
This large magnetic field creates a significant force on a small length of wire. As the angle of the magnetic field becomes more closely aligned to the current in the wire, there is less of a force on it, as seen from comparing parts a and b.
Exercise \(\PageIndex{1}\)
A straight, flexible length of copper wire is immersed in a magnetic field that is directed into the page. (a) If the wire’s current runs in the + x -direction, which way will the wire bend? (b) Which way will the wire bend if the current runs in the – x -direction?
Solution
a. bends upward; b. bends downward
Example \(\PageIndex{3}\): Force on a Circular Wire
A circular current loop of radius R carrying a current I is placed in the xy -plane. A constant uniform magnetic field cuts through the loop parallel to the y -axis (Figure \(\PageIndex{4}\)). Find the magnetic force on the upper half of the loop, the lower half of the loop, and the total force on the loop.
Strategy
The magnetic force on the upper loop should be written in terms of the differential force acting on each segment of the loop. If we integrate over each differential piece, we solve for the overall force on that section of the loop. The force on the lower loop is found in a similar manner, and the total force is the addition of these two forces.
Solution
A differential force on an arbitrary piece of wire located on the upper ring is:
\[dF = I B \, sin \, \theta \, dl, \nonumber \] where \(\theta\) is the angle between the magnetic field direction (+ y ) and the segment of wire. A differential segment is located at the same radius, so using an arc-length formula, we have:
\[dl = Rd\theta \nonumber \]
\[dF = IBR \, sin \, \theta \, d\theta. \nonumber \]
In order to find the force on a segment, we integrate over the upper half of the circle, from 0 to \(\pi\). This results in:
\[F = IBR \int_0^{\pi} sin \, \theta \, d\theta = IBR(-cos \pi + cos 0) = 2 IBR. \nonumber \]
The lower half of the loop is integrated from \(\pi\) to zero, giving us:
\[F = IBR \int_{\pi}^0 sin \, \theta \, d\theta = IBR(-cos 0 + cos \pi) = -2 IBR. \nonumber \]
The net force is the sum of these forces, which is zero.
Significance
The total force on any closed loop in a uniform magnetic field is zero. Even though each piece of the loop has a force acting on it, the net force on the system is zero. (Note that there is a net torque on the loop, which we consider in the next section.)


2.4G: Preparing the Research Report
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Sociological research publications generally include a literature review, an overview of the methodology followed, the results and an analysis of those results, and conclusions.
Learning Objectives
- Describe the main components of a sociological research paper
Key Points
- Like any research paper, a sociological research report typically consists of a literature review, an overview of the methods used in data collection, and analysis, findings, and conclusions.
- A literature review is a creative way of organizing what has been written about a topic by scholars and researchers.
- The methods section is necessary to demonstrate how the study was conducted, including the population, sample frame, sample method, sample size, data collection method, and data processing and analysis.
- In the findings and conclusion sections, the researcher reviews all significant findings, notes and discusses all shortcomings, and suggests future research.
Key Terms
- methodology : A collection of methods, practices, procedures, and rules used by those who work in some field.
- quantitative : Of a measurement based on some quantity or number rather than on some quality.
- literature review : A literature review is a body of text that aims to review the critical points of current knowledge including substantive findings as well as theoretical and methodological contributions to a particular topic.
Like any research paper, sociological research is presented with a literature review, an overview of the methods used in data collection, and analysis, findings, and conclusions. Quantitative research papers are usually highly formulaic, with a clear introduction (including presentation of the problem and literature review); sampling and methods; results; discussion and conclusion. In striving to be as objective as possible in order to reduce biased interpretations of results, sociological esearch papers follow the scientific method. Research reports may be published as books or journal articles, given directly to a client, or presented at professional meetings.
A literature review is a creative way of organizing what has been written about a topic by scholars and researchers. You will find literature reviews at the beginning of many essays, research reports, or theses. In writing the literature review, your purpose is to convey to your reader what you have learned through a careful reading of a set of articles related to your research question.
A strong literature review has the following properties:
- It is organized around issues, themes, factors, or variables that are related directly to your thesis or research question.
- It shows the path of prior research and how the current project is linked.
- It provides a good synthesis of what is, and is not, known.
- It indicates the theoretical framework with which you are working.
- It identifies areas of controversy and debate, or limitations in the literature sharing different perspectives.
- It places the formation of research questions in their historical context.
- It identifies the list of the authors that are engaged in similar work.
The methodssection is necessary to demonstrate how the study was conducted, and that the data is valid for study. Without assurance that the research is based on sound methods, readers cannot countenance any conclusions the researcher proposes. In the methodology section, be sure to include: the population, sample frame, sample method, sample size, data collection method, and data processing and analysis. This is also a section in which to clearly present information in table and graph form.
In the findings and conclusion sections, the researcher reviews all significant findings, notes and discusses all shortcomings, and suggests future research. The conclusion section is the only section where opinions can be expressed and persuasive writing is tolerated.
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- OpenStax College, Introduction to Sociology. July 3, 2012. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m42796/latest/?collection=col11407/latest . License : CC BY: Attribution
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- OpenStax College, Introduction to Sociology. July 3, 2012. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m42796/latest/?collection=col11407/latest . License : CC BY: Attribution
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- Help question coding - help desk. Provided by : Wikimedia. Located at : commons.wikimedia.org/wiki/File:Help_question_coding_-_help_desk.PNG . License : Public Domain: No Known Copyright
- Correlation, Causation, and Spurious Relationships. Located at : http://www.youtube.com/watch?v=UNonyq1yhiE . License : Public Domain: No Known Copyright . License Terms : Standard YouTube license
- Linear regression. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Linear_regression.svg . License : Public Domain: No Known Copyright
- OpenStax College, Introduction to Sociology. July 3, 2012. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m42796/latest/?collection=col11407/latest . License : CC BY: Attribution